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Hyperbolic and Spherical Tiling Gallery
{{Tessellation Hyperbolic tilings Infinite-order tilings Isogonal tilings Isohedral tilings Pentagonal tilings Regular tilings
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ...
, the infinite-order pentagonal tiling is a regular
Regular may refer to:
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings
Other uses
* Regular character, ...
tiling. It has Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
of . All vertices are ''ideal'', located at "infinity", seen on the boundary of the Poincaré hyperbolic disk projection.
Symmetry
There is a half symmetry form, , seen with alternating colors: :
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (5n).See also
*Pentagonal tiling
In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon.
A regular tiling, regular pentagonal tiling on the Euclidean plane is impossible because the internal angle of a Pentagon#Regular ...
*Uniform tilings in hyperbolic plane
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as Face (geometry), faces and is vertex-transitive (Tran ...
*List of regular polytopes
This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces.
Overview
This table shows a summary of regular polytope counts by rank.
There are no Euclidean regular star tessellations in any number of dimensions.
...
References
* *External links
* *Hyperbolic and Spherical Tiling Gallery
{{Tessellation Hyperbolic tilings Infinite-order tilings Isogonal tilings Isohedral tilings Pentagonal tilings Regular tilings